A Recursive Construction of the Regular
Exceptional Graphs with Least Eigenvalue −2
I. Barbedo, D. M. Cardoso, D. Cvetkovi´
c, P. Rama, S.K. Simi´
c
∗Abstract.
In spectral graph theory a graph with least eigenvalue −2 is exceptional if it is con-nected, has least eigenvalue greater than or equal to −2, and it is not a generalized line graph. A (κ, τ )-regular set S of a graph is a vertex subset, inducing a κ-regular subgraph such that every vertex not in S has τ neighbors in S. We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.
Mathematics Subject Classification (2010). Primary 05C50; Secondary 06A06. Keywords. Spectral graph theory, exceptional graphs, posets.
1. Introduction
Let G = (V (G), E(G)) be a simple graph, where V (G) denotes the nonempty set of vertices and E(G) the set of edges. It is assumed that G is of order n, i.e. |V (G)| = n. An edge of E(G), which has the vertices i and j as end-vertices is denoted by ij. When there is an edge between the vertices i and j we say that these vertices are adjacent. The neighborhood of a vertex u ∈ V (G), NG(u), is
the set of vertices adjacent to u, that is, NG(u) = {v ∈ V (G) : uv ∈ E(G)}. The
degree of vertex u is the cardinality of its neighborhood. A graph G is r-regular (or regular of degree r) if each vertex of G has the same degree r.
Throughout the paper, AG denotes the adjacency matrix of G, that is AG =
(aij)n×n, where aij = 1 if ij ∈ E(G) and aij = 0 otherwise. The eigenvalues of
the graph G are the eigenvalues of its adjacency matrix, here ordered such that λ1 ≥ λ2 ≥ · · · ≥ λn. A detailed treatment of graph eigenvalues can be found in
[8].
∗The authors I. Barbedo, D. M. Cardoso and P. Rama were partially supported by Portuguese
funds through the CIDMA - Center for Research and Development in Mathematics and Appli-cations, and the Portuguese Foundation for Science and Technology (“FCT-Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.
A pair (X, B) is a 1 − (v, l, λ) design, if X is a set with cardinality v and B is a family of b subsets of X with cardinality l, called blocks, and each element x ∈ X lies in exactly λ blocks. The incidence matrix B of a 1 − (v, l, λ) design is the v × b matrix with ij-entry equal to 1 if xi∈ Bj and 0 otherwise. Alternatively,
a 1 − (v, l, λ) design (X, B) can be represented by a semi-regular bipartite graph with parameters (v, b, λ, l), i.e. by a bipartite graph with v vertices of degree λ in one colour class, and b vertices of degree l in another colour class.
A connected graph with least eigenvalue greater than or equal to −2 is either a generalized line graph (with line graphs included), or an exceptional graph (see, e.g., [11]). According to [11, Proposition 1.1.9], a regular connected generalized line graph (see definition, e.g., in [11, Section 1.1]) is either a line graph or a cocktail party graph (a regular graph on 2k vertices of degree 2k − 2). A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to −2, and it is not a generalized line graph. It is known [10] that an exceptional graph has at most 36 vertices, with vertex degrees at most 28. There are exactly 187 regular exceptional graphs. They are divided into three subsets (to be defined later) called layers. These graphs are constructed in [11, 1, 2] using different techniques. A comprehensive survey of problems on graphs with least eigenvalue at least −2, including exceptional graphs, can be found in [11].
A vertex subset S ⊆ V (G) of the graph G is a stable (or independent) set if no pair of vertices in S is connected by an edge. A stable set with maximum cardinality is a maximum stable set.
Given a graph G, a partition π = (V1, . . . , Vr) of the vertex set of G is an equitable partition, if for any pair i, j ∈ {1, . . . , r} there exists dij ∈ N0 such that
for all v ∈ Vi dij= |NG(v) ∩ Vj|, that is, the number of neighbors that a vertex of
Vi has in Vj is independent of the choice of the vertex in Vi.
A (κ, τ )-regular set S of a graph G is a vertex subset which induces a κ-regular subgraph such that every vertex not in S has τ neighbors in S. If G is a regular graph, then a (κ, τ )-regular set S defines an equitable bipartition in G.
The (κ, τ )-regular sets appeared first in [16], under the designation of eigen-graphs, and also in [15], in both cases in the context of strongly regular graphs and designs. Later on, the (κ, τ )-regular sets were investigated in the general context of arbitrary graphs [4, 6, 5].
The aim of this paper is to present a recursive construction of all regular ex-ceptional graphs based on the new (κ, τ )-extension technique suggested in [3]. It recursively generates the families of regular exceptional graphs along with a partial order relation among them, and this is represented by its Hasse diagram.
In Section 2 we describe the (κ, τ )-extension technique of a regular graph G by a regular graph H, and also the partial order relation that arises. The process of extending a graph is reduced to the construction of the incidence matrices of a 1-design (or an appropriate bipartite semi-regular graph).
In Section 3 we propose a technique to construct the regular exceptional graphs, by (0, 2)-extensions in the case of 1st and 2nd layer, and by (1, 3)-extensions in the case of 3rd layer. In each case we describe how to construct the incidence matrices
of the 1-design desired, stating some proprieties. In addition, we also propose an algorithm to construct the regular exceptional graphs in each layer.
In Section 4 we describe, in more details, the computational results that were obtained by the algorithm for the three layers.
The Appendix contains for each regular exceptional graph the list of other reg-ular exceptional graphs with a minimal number of vertices in which it is contained as a proper induced subgraph.
2. Construction of regular graphs by (κ, τ )-extensions
Let G be a (p − τ )-regular graph of order n1(with τ > 0) and H a κ-regular graph
(with κ < p) of order n2. Our aim is to obtain a p-regular graph H ⊕ G, of order
n2+ n1, such that each vertex in G has τ neighbors in H, and each vertex in H has
exactly p − κ neighbors in G (hence V (H) is a (κ, τ )-regular set in H ⊕ G). The procedure that generates the graph H ⊕ G from G is called a (κ, τ )-extension of G
of size n2. This construction is possible if we can define a family, S, of n1subsets
in V (H), called blocks, each of them with cardinality τ , so that each v ∈ V (H) is in exactly p − k blocks of S, that is, (V (H), S) is a 1 − (n2, τ, p − k) design.
Note that there is a 1 − (n2, τ, p − k) design if and only if nn1
2 =
p−κ
τ . Hence the
adjacency matrix of H ⊕ G is given by AH B BT A G ,
where AH and AG are the adjacency matrices of H and G, respectively, and B
is the incidence matrix of a 1 − (n2, τ, p − κ) design, that is, each column of B
is the characteristic vector of a block. Further on, we shall use graph theoretical terminology and consider the corresponding semi-regular bipartite graphs.
The (κ, τ )-extension of a regular graph G by a regular graph H to obtain another regular graph H ⊕ G, can be applied recursively to generate a sequence of regular graphs. Considering the (p − τ )-regular graph G and the κ-regular graph H described above, starting with G0 = G, we can generate a set F of
((p − τ ) + mτ )-regular graphs, Gm, of order n1+ m n2, where each Gmis obtained
by a (κ, τ )-extension of Gm−1 (m ≥ 1). Consequently, we can define the following
partial order relation on F, or on any set of graphs. Definition 2.1. Given κ, τ, s, if G, G′
are regular graphs, then G G′
if and only if G′
can be obtained from G by a sequence of zero or more (κ, τ )-extensions of size s.
3. Construction of regular exceptional graphs
The set of regular exceptional graphs is partitioned into three layers according to degree r and order n [11, Theorem 4.1.5]. Any such graph is in the 1st, 2nd or 3rd
layer, if the following holds, respectively: (i) n = 2(r + 2) ≤ 28,
(ii) n = 3
2(r + 2) ≤ 27 and G is an induced subgraph of the Schl¨afli graph,
(iii) n = 4
3(r + 2) ≤ 16 and G is an induced subgraph of the Clebsch graph.
There are 163 graphs in the 1st layer, 21 in the 2nd layer and 3 in the 3rd layer, i.e. 187 in total.
The regular exceptional graphs are completely described in Table A3 in [11, pp. 213-227]. As in [11], regular exceptional graphs are denoted by numbers 1 − 187.
Let L be the set of regular graphs whose least eigenvalue is greater than or equal to -2. Hence L includes all regular exceptional graphs. We shall consider subsets L1, L2, L3 of L in which the ratio r+2n (n the number of vertices, r the
degree) is the same as in layers 1, 2, 3, respectively. Hence, each layer is a subset of the corresponding set L1, L2 or L3.
Let G be a regular graph of degree r and order n. Let α(G) be the size of a maximum stable (or independent) set of G and λn its least eigenvalue. Then
α(G) ≤ −nλn r − λn
.
This inequality is known as the Hoffman inequality although A.J. Hoffman never published it. For some bibliographical details related to this bound see [3].
In [3] it is noted that in the case of regular exceptional graphs of the 1st and 2nd layers, the Hoffman upper bound is attained and is equal to the cardinality of a maximum stable set, which is 4 and 3, respectively. This observation has an empirical character and is based on an inspection of the regular exceptional graphs. In fact, the Hoffman bound is attained for a regular graph G if and only if G has a (0, τ )-regular set such that τ = −λn. The necessary condition was proved
in [14] (see also [13, Lemma 9.6.2]) and the sufficient condition was proved in [7]. The following theorem stems from [12].
Theorem 3.1. Let G be an r-regular graph and λn= −τ its least eigenvalue. For any stable set S of size s and characteristic vector z, we have:
s ≤ nτ r + τ.
Furthermore, the following are equivalent: (i) Equality holds.
(ii) z is a linear combination of a r-eigenvector and a λn-eigenvector.
(iii) The bipartite subgraph induced by the partition {S, V (G) \ S} is semi-regular.
It is easy to prove that the stability number of regular exceptional graphs from the first layer is 3 or 4, in the 2nd layer is 2 or 3, and in the 3rd layer is 2. In fact, if S is a maximum stable set of an r-regular graph G, then
n − α(G) = [ i∈S NG(i) ≤X i∈S |NG(i)| = α(G)r ⇒ α(G) ≥ n r + 1.
Therefore, taking in account the relation between the order n of the regular ex-ceptional graphs and their regularities in each layer, combining the Hoffman upper bound with the above lower bound, the result follows.
However, in spite of some efforts, we were not able to prove theoretically that the Hoffman upper bound is attained in regular exceptional graphs in the 1st and 2nd layer.
We shall construct all regular exceptional graphs from the first and second layer by (0, 2)-extensions. For the first layer the (0, 2)-extension will have size 4 and for the second layer the size is 3. Applications of (0, 2)-extensions will create a stable set of the corresponding size in the resulting graph. This stable set will be a maximum stable set by the Hoffman bound and, from this construction, we may conclude that the Hoffman upper bound is attained for all regular exceptional graphs in the 1st and 2nd layer.
The graphs from the 3rd layer will be considered separately and they will be built by (1, 3)-extensions.
With each type of (κ, τ )-extension we consider the corresponding partial order relation .
As observed in [3], we may state the following result.
Theorem 3.2. Regular exceptional graphs are not minimal elements of the posets (L1, ) and (L2, ).
As we shall see the same holds for (L3, ).
Together with L1, L2, L3 one can study the posets E1, E2, E3 of exceptional
graphs from the three layers with corresponding relations . It would be interesting to study the structure of all these posets. The sets L1, L2, L3 are infinite while
the sets E1, E2, E3 contains 163, 21, 3 elements respectively.
Throughout the recursive construction, starting from a regular exceptional graph G1, with least eigenvalue −2, every regular graph Gi, with i ≥ 2, obtained
by (κ, τ )-extensions and with least eigenvalue −2, remains an exceptional graph. This conclusion is a consequence of the fact that the graph property of being a line graph is a hereditary property. This means that any induced subgraph of a line graph is also a line graph. Hence we have the following proposition.
Proposition 3.3. Let G be a regular graph with least eigenvalue −2. Let H be a
regular induced subgraph of G. We have
(i) if G is a line graph then H is also a line graph,
3.1. Construction of regular exceptional graphs in the 1st layer by (0, 2)-extensions of size 4. The smallest regular exceptional graphs in the first layer are the five graphs Z1, Z2, Z3, Z4and Z5of order 10 and degree 3, given in Figure
1 (taken from [11, Appendix A.3]).
s s s s s s s s s s ❅ ❅ ❅❅ ❆ ❆ ✁✁ ✁ ✁ ❆❆ ❆ ❆ ✁✁ ✁ ✁ ❆❆ Z1 s s s s s s s s s s ❅ ❅ ✁✁ ✁✁✁ ❆ ❆ ❆ ❆ ❆ ❅❅ ❆ ❆ ✁✁ Z2 s s s s s s s s s s P P P ✏✏✏ ✑ ✑ ✑ ◗◗◗ ❆ ❆ ✁✁ ❆ ✁✁❆ ✂ ✂✂ ❜ ❜ ❜❜ ❇❇❇ ✧ ✧ ✧ ✧ Z3 s s s s s s s s s s ❍❍ ✟✟ ✟ ✟❍ ❍ ☞☞ ☞☞ ▲ ▲ ▲ ▲ ✟✟ ❍ ❍✟ ✟ ❍❍ Z4 s s s s s s s s s s ❈ ❈ ❈ ✄✄ ✄ ❅ ❅❅ ✂ ✂✂❇❇❇ P P ❜❜✧✧✏✏ ✁✁ ❆❆ Z5
Figure 1. The smallest regular exceptional graphs in the 1st layer.
These graphs are obtained by (0, 2)-extensions of the graph 3K2of order 6 and
regularity 1, which is a line graph.
The 8 exceptional graphs of order 12 and regularity 4, are obtained by (0, 2)-extensions of graphs of order 8 and regularity 2, that is, one of the three graphs 2C4, C3˙∪C5(disjoint union of graphs C3and C5) or C8, which are also line graphs.
Since for regular exceptional graphs of the first layer r = n
2 − 2, with 10 ≤
n ≤ 28, in order to build the Hasse diagram of the first layer partially ordered by the relation , this set of exceptional graphs is divided into two partially ordered subsets: the graphs with even regularity, obtained by (0, 2)-extensions from a graph with even regularity, and the graphs with odd regularity, obtained by (0, 2)-extensions from a graph with odd regularity.
If G is a graph of even order n and regularity r = n
2− 2 and H is the 0-regular
graph of order 4, the adjacency matrix of the graph G′ of order n′ = 4 + n and
regularity r′ =4+n 2 − 2 =
n
2 obtained from a graph G by a (0, 2)-extension is given
by AG′ = O4 B BT A G ,
where O4is the adjacency matrix of the graph H, that is, the null square matrix
of order 4, AG is the adjacency matrix of G and B is a matrix with 4 rows and n
Assuming as a known fact that in the case of regular exceptional graphs of the 1st and 2nd layers, the Hoffman upper bound is attained all these graphs can be constructed by extending graphs with additional 4 or 3 vertices in the way implied by the above considerations.
Let us describe (0, 2)-extensions of size 4 in some detail.
The graph G is extended by 4 vertices which form a set S = V (H). Let 1, 2, 3, 4 be the vertices of S. Each vertex of G should become adjacent to exactly two vertices of S. Let us define an r-regular multigraph M (S) having the set S as the vertex set. If a vertex v of G becomes adjacent to vertices x, y of S, then there is an edge labelled v between x and y in M (S). In this way, the vertices of G subdivide the edges of M (S). There are six 2-element subsets of S. Constructing G′ from G by an (0, 2)-extension means, in fact, to partition the vertex set of G
into six subsets which, in turn, should be assigned to 2-element subsets of S in such a way that M (S) is regular of degree r. However, the resulting graph G′need
not to be an L-graph, and this should be checked in the actual constructions. A multigraph M (S) can be associated with a weighted complete graph on the four vertices of S (see Fig. 2). The weight xi on the i-th edge of the multigraph
M (S) represents the number of vertices of G adjacent to corresponding vertices in S. t t t t ✚✚ ✚✚ ✚✚ ✁✁ ✁✁ ✁✁ ✁✁ ✁✁ ✁✁ ❩ ❩ ❩ ❩ ❩ ❩ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ 1 2 4 3 x1 x6 x5 x2 x3 x4
Figure 2. A weighted complete graph with vertex set {1, 2, 3, 4}.
We have x1+ x2+ x3 = n/2 since this is the degree of 1 in M (S). Hence,
x4+ x5+ x6= n/2. We can conclude that the sum of weights of the edges of any
star K1,3 and of any triangle K3 is equal to n/2. In addition, we have x1 = x6,
x2= x5 and x3= x4.
Because of automorphisms it is sufficient to consider partitions of the vertex set of G determined by the weights x1, x2, x3 with 0 ≤ x1≤ x2≤ x3≤ n/2.
Now we shall propose an algorithm to construct the 163 graphs of the first layer by (0, 2)-extensions which produce equitable partitions.
For a given n construct all feasible triplets (x1, x2, x3). For each triplet (x1, x2, x3)
we find in turn all ordered partitions of the vertex set of G into parts of cardinalities x1, x2, . . . , x6. For each such partition we consider all graphs G and extend them
to a graph G′ according to this partition. Then, for every graph G′ we calculate
the least eigenvalue and if it is equal to −2 we have generated a regular exceptional graph on n + 4 vertices. Finally, we should eliminate isomorphic duplicates but record all graph pairs which are in relation .
A suitable computer program based on the above procedure has generated all 163 regular exceptional graph in the first layer. The results are given in the Appendix where for each graph the list of its immediate successors in the poset L is given.
3.2. Construction of regular exceptional graphs in the 2nd layer by (0, 2extensions of size 3. Using the procedure of construction by (κ, τ )-extensions described in Section 2, our aim is now to build a set of regular graphs obtained by a (0, 2)-extension, H ⊕ G, such that G is a r-regular graph of order n, with r = 2n
3 − 2, and H is the 0-regular graph of order |V (H)| = 3. Therefore,
this set includes the regular exceptional graphs of the 2nd layer.
The regular exceptional graphs of the 2nd layer of the lowest order are the graphs of order 9 and regularity 4 in Fig. 3. These graphs are obtained by a (0, 2)-extension of the 2-regular graph C6 and the disconnected graph 2K3, which are
line graphs. To construct the Hasse diagram that represents the partially ordered set E2 with relation of the regular exceptional graphs in the second layer, we
consider the r-regular graphs of order n, with r = 2n
3 − 2 and 6 ≤ n ≤ 27. t t t t t t t t t ◗ ◗ ◗ ❆ ❆ ✁✁ ✑ ✑ ✑ ✁✁❆❅❆❅ ❅ ❅ ❅ ✟✟❍❍ t t t t t t t t t ❆ ❆ ❆ ❆ ✁✁ ✁✁ ✟✟✟✟ ✟✟ ✁✁ ✁✁ ❆ ❆ ❆ ❆ ❍ ❍ ❍ ❍ ❍ ❍ ✦ ✦ ✦ ✦ ✦ ✁✁❛❛❆❆ ❛❛❛ ❍ ❍ ✟✟ ❍ ✟✟❍
Figure 3. The smallest regular exceptional graphs in the 2nd layer.
In fact, in order to construct each adjacency matrix of the graph G′, obtained
from G by a (0, 2)-extension, G′ = H ⊕G, it is necessary to determine the incidence
matrix with 3 rows and n columns.
Thus, in order to construct the graphs of the second layer we developed an algorithm similar to the algorithm described in the case of graphs from the first layer. The results are given in Appendix. The Hasse diagram of graphs from the second layer is given in Fig. 4.
t C6 t2K3 t164 t 165 t 166 t 170 t 167 t 169 t 168 t 174 t 176 t172 t173 t171 t 175 t 179 t180 t177 t178 t182 t181 t183 t184 ❩ ❩ ❩ ❩ ✚✚ ✚✚ ❩ ❩ ❩ ❩ ✚✚ ✚✚ ▲ ▲ ▲ ▲ ▲ ☞☞ ☞☞ ☞ ▲ ▲ ▲ ▲ ▲ ✱✱ ✱✱ ✱✱ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ❧ ▲ ▲ ▲ ▲ ▲ ☞☞ ☞☞ ☞ ✱✱ ✱✱ ✱✱ ▲ ▲ ▲ ▲ ▲ ❧ ❧ ❧ ❧ ❧ ❧ ▲ ▲ ▲ ▲ ▲ ☞☞ ☞☞ ☞ ✁✁ ✁✁ ✑✑ ✑✑ ✑✑ ✁✁ ✁✁ ✑✑ ✑✑ ✑✑ ◗ ◗ ◗ ◗ ◗ ◗ ✁✁ ✁✁ ✑✑ ✑✑ ✑✑ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✑ ✑ ✑ ✑ ✑ ✑ ✁ ✁ ✁ ✁ ❛❛ ❛❛ ❛❛ ❛❛ ❛❛ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆
Figure 4. Hasse diagram of graphs from the 2nd layer.
3.3. Construction of regular exceptional graphs in the 3rd layer by (1, 3)-extensions of size 4. The exceptional graphs from the 3rd layer are only three. The Hoffman bound is not attained for these graphs. However, it is also possible to build them using the procedure of construction by (κ, τ )-extensions described in Section 2, in this case, by (1, 3)-extensions of size 4.
So, the aim now is to build a set of regular graphs obtained by an (1, 3)-extension, H ⊕ G, such that G is a r-regular graph of order n, with r = 4n
3 − 2,
and H is the 1-regular graph of order 4,i.e. 2K2.
Starting from the line graph 2K2 and extending the graphs in turn by four
vertex (1,3)-regular sets we obtained the exceptional graphs 185, 186 and 187. They have 8, 12 and 16 vertices respectively. Graph 185 is presented in Fig. 5. Graph 187 is the well known Clebsh graph.
4. Computational results
Our algorithm was implemented in Matlab R2009b in order to construct the ad-jacency matrices of the regular exceptional graphs of the first layer. Similar algo-rithms were implemented in the cases of the graphs of the second and third layers. The results produced in each layer are described in the Appendix.
t t t t t t t t ✟ ✟ ✟ ✟ ❍❍❍❍ ❍ ❍ ❍ ❍ ✟✟✟✟ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ✁✁ ✁✁ ❅ ❅ ❅ ❅
Figure 5. The smallest regular exceptional graph in the 3rd layer.
has four components. Two of these components are associated with the first layer of the regular exceptional graphs: one for the graphs with even degree and the other for the graphs with odd degree.
Notice that there are 3 regular exceptional graphs from the first layer that are not obtained by a (0, 2)-extension of the minimal graphs 3K2, 2C4, C3˙∪C5, C8:
the 5-regular exceptional graph 17, which is obtained by a (0, 2)-extension from the disconnected line graph L1 (see Fig. 6), the 6-regular exceptional graph 56,
which is obtained by (0, 2)-extension from the 4-regular line graph L2 (see Fig.
6) and the 8-regular exceptional graph 134, which is obtained by (0, 2)-extension from the 6-regular line graph L3(see the root graph of L3in Fig. 7)
t t t t t t t t t t ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ L1 t t t t t t t t t t t t ✁✁ ✁✁ ❆ ❆ ❆❆ ❅ ❅ ❅ ❅ ✁✁ ✁✁ ❅ ❅ ❆ ❆ ❆❆ L2
Figure 6. The line graphs L1 and L2.
The maximal elements in the first layer are the eight regular exceptional graphs of order 20 (113-117, 119-121), all the regular exceptional graphs with order 22 (135-152), and the three Chang graphs (strongly regular exceptional graphs of order 28).
The third component of the Hasse diagram is associated with the regular ex-ceptional graphs from the second layer (see Fig. 4). The minimal elements are C6 and 2K3 and the maximal elements are the regular exceptional graphs 181 and
184 (the Schl¨afli graph).
For the third layer there is a minimal graph 2K2and a strongly regular graph,
t t t t t t t t ◗ ◗ ◗ ◗ ◗ ◗ ✑✑ ✑✑ ✑✑ ❆ ❆ ❆ ❆ ✁✁ ✁✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆❆ ◗ ◗ ◗ ◗ ◗◗ ✁ ✁ ✁ ✁ ❆ ❆ ❆❆ ✑ ✑ ✑ ✑ ✑ ✑ ❅ ❅ ❅ ❅ ❅❅ ✟ ✟ ✟ ✟ ❍❍❍❍
Figure 7. The root graph of the line graph L3.
5. References
References
[1] F.C. Bussemaker, D. Cvetkovi´c, and J.J. Seidel, Graphs related to exceptional root systems. T.H.-Report 76-WSK-05, Technological University Eindhoven, 1976. [2] F.C. Bussemaker, D. Cvetkovi´c, and J.J. Seidel, Graphs related to exceptional
root systems. In Combinatorics, Proc. V Hungarian Colloquium on Combinatorics, Keszthelly 1976 (ed. A. Hajnal, V. T. S´os), Vol. I, Amsterdam-Oxford-New York 1978, 185-191.
[3] D. M. Cardoso and D. Cvetkovi´c, Graphs with least eigenvalue -2 attaining a convex quadratic upper bound for stability number, Bull. Acad. Serbe Sci. Arts, Cl. Sci. Math. Natur., Sci. Math.133(31) (2006), 41–55.
[4] D. M. Cardoso and P. Rama, Equitable bipartitions of graphs and related results, J. Mathematical Sciences120(1) (2004), 869–880.
[5] D. M. Cardoso and P. Rama, Spectral results on graphs with regularity constraints, Linear Algebra Appl.423(1) (2007), 90–98.
[6] D. M. Cardoso and P. Rama, Spectral results on regular graphs with (κ, τ )-regular sets, Discrete Math. 307(11-12) (2007), 1306–1316.
[7] D. M. Cardoso, M. Kaminski, and V. Lozin, Maximum k-regular induced subgraphs, J. Comb. Optim.14(4) (2007), 455–463.
[8] D. Cvetkovi´c, M. Doob, and H. Sachs, Spectra of Graphs - Theory and Application. Academic Press, New York, 1980.
[9] D. Cvetkovi´c, P. Rowlinson, and S. Simi´c, Graphs with least eigenvalue -2: The star complement technique, J. Algebraic Combinatorics 14(1) (2001), 5–16.
[10] D. Cvetkovi´c, M. Lepovi´c, P. Rowlinson, and S. Simi´c, The maximal exceptional graphs, J. Combinatorial Theory Ser. B, 86(2) (2002), 347–363.
[11] D. Cvetkovi´c, P. Rowlinson, and S. Simi´c, Spectral Generalizations of Line Graphs, on Graphs with Least Eigenvalue -2. Cambridge University Press, Cambridge, 2004. [12] C.D. Godsil and M.W. Newman, Eigenvalue bounds for independent sets,
[13] C. D. Godsil and G. Royle, Algebraic Graph Theory. Graduate texts in mathematics. Springer-Verlag, New York, 2001.
[14] W. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226/228 (1995), 593–616.
F.C. Bussemaker, D. Cvetkovi´c, and J.J. Seidel, Graphs related to exceptional root systems. In Combinatorics, Proc. V Hungarian Colloquium on Combinatorics, Keszthelly 1976 (ed. A. Hajnal, V. T. S´os), Vol. I, Amsterdam-Oxford-New York 1978, 185-191.
[15] A. Neumaier, Regular sets and quasi-symmetric 2-designs. In Combinatorial Theory (ed. D. Jungnickel and K. Vedder), Lecture Notes in Mathematics, Vol. 969, Springer, Berlin - Heidelberg 1982, 258–275.
[16] D. M. Thompson, Eigengraphs: constructing strongly regular graphs with block de-signs, Utilitas Math. 20 (1981), 83–115.
APPENDIX: Extensions of regular exceptional graphs
The data on the 187 regular exceptional graphs are given in Table A3 from the book [11]. These graphs are denoted here by numbers 1 – 187 and these numbers refer to [11]. The graphs are divided into three layers and into smaller groups according to the number of vertices n and the degree r. For each graph the list of regular exceptional graphs obtained by (κ, τ )-extensions is given.
First layer n = 10, r = 3 1. 14, 16, 18, 22, 23, 24, 27, 28, 30, 32, 33 2. 19, 20, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 34 3. 18, 27, 28, 29, 30, 32, 33, 34 4. 15, 16, 18, 19, 20, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 5. 30 n = 12, r = 4 6. 37, 49, 51, 57, 58, 68 7. 39, 40, 41, 42, 46, 47, 48, 51, 52, 53, 55, 57, 58 8. 39, 40, 42, 55, 57, 61, 63, 64, 65, 66, 67, 68 9. 35, 37, 44, 45, 59, 60, 69 10. 35, 36, 40, 41, 42, 47, 48, 49, 53, 60, 62, 64, 65 11. 35, 37, 38, 39, 40, 46, 47, 48, 49, 50, 51, 54, 59, 60, 62, 63, 64, 67, 68 12. 39, 40, 46, 47, 48, 49, 50, 51, 52, 54, 57, 58, 61, 62, 63, 64 13. 43, 44, 45, 46, 49, 51, 59, 60, 67 n = 14, r = 5 14. 70, 83, 84, 88, 89, 97, 98, 99, 100 15. 81, 82, 90, 95, 96, 105 16. 84, 85, 91, 92, 93, 94, 97, 107 17. 89, 100 18. 90, 92, 94, 97, 99, 101, 103 19. 71, 72, 73, 74, 77, 78, 86, 87, 88, 90, 91, 94, 95 20. 71, 72, 74, 78, 79, 86, 88, 89, 91, 92, 93, 94 21. 78, 88, 93 22. 71, 72, 74, 75, 76, 80, 96, 97, 99, 100, 103, 104 23. 71, 74, 76, 77, 78, 79, 97, 98, 99, 101, 102, 104 24. 71, 72, 75, 76, 81, 84, 91, 92, 94, 100 103, 104 25. 72, 73, 77, 84, 87, 88, 89, 97, 98, 99 26. 71, 77, 78, 82, 83, 86, 91, 97, 102, 104 27. 71, 72, 77, 78, 79, 82, 84, 85, 90, 93, 94, 95, 96, 97, 101, 102, 103 28. 71, 72, 74, 75, 76, 78, 79, 83, 84, 86, 88, 92, 93, 96, 97, 102, 103 29. 86, 90, 94, 106, 107 30. 88, 92, 94, 95, 105, 107
31. 77, 78, 82, 86, 87, 93, 97, 101, 102, 107 32. 71, 72, 77, 79, 82, 85, 90, 91, 92, 93, 96, 97, 101, 102, 103, 105, 107 33. 71, 72, 75, 76, 78, 79, 83, 84, 86, 88, 89, 91, 94, 97, 98, 99, 100, 102, 103, 104, 106, 107 34. 74, 78, 84, 91, 94, 95, 99, 102, 103, 107 n = 16, r = 6 35. 108, 109, 111, 123, 124, 126, 128, 129, 132 36. 108, 121, 128, 131 37. 110, 111, 112, 123, 124, 129, 130, 133 38. 120, 124, 128 39. 115, 117, 118, 119, 120, 121, 122, 123, 124, 126, 130, 131 40. 116, 117, 119, 121, 123, 124, 125, 126, 127, 128, 129, 133 41. 113, 116, 121, 123, 126, 127 42. 114, 116, 117, 120, 121, 125, 128, 131, 132 43. 108, 109, 122, 130 44. 109, 112, 127, 129 45. 109, 111, 124, 132, 133 46. 113, 114, 115, 116, 118, 119, 122, 124, 126, 127, 128 47. 113, 116, 117, 120, 121, 124, 126, 128, 131 48. 114, 117, 120, 121, 128 49. 113, 116, 119, 123, 124, 127, 129, 133 50. 113, 114, 124, 127, 129 51. 114, 115, 116, 118, 123, 124, 127, 129, 130 52. 118, 119, 121, 122, 125 53. 116, 117, 121, 125, 131 54. 114, 118, 120, 125, 131 55. 114, 116, 117, 120 56. 118 57. 116, 117, 118 58. 113, 114, 115, 116, 117, 118, 119 59. 109, 112, 124, 129, 130, 133 60. 108, 111, 126, 127, 128, 132, 133 61. 114, 117, 118, 119 62. 113, 114, 117, 119, 120 63. 115, 116, 118, 119 64. 116, 117, 120 65. 128 66. 131 67. 127, 130, 132 68. 129 69. 111, 112 n = 18, r = 7
70. 142, 143, 146 71. 136, 138, 139, 140, 141, 142 72. 136, 138, 140, 143, 144 73. 135, 136, 142, 145 74. 138, 139, 144, 145 75. 136, 142, 146, 147 76. 136, 138, 143, 147 77. 137, 138, 139, 141 78. 139, 141, 142, 143 79. 138, 141, 144 80. 136, 146 81. 136, 140, 149 82. 137, 140, 141, 152 83. 138, 142, 144, 147, 151 84. 138, 139, 140, 141, 142, 143, 144, 145, 150, 151 85. 140, 141, 152 86. 138, 149, 151 87. 139, 150 88. 142, 150, 151 89. 143, 151 90. 137, 140, 148, 149, 150, 152 91. 141, 144, 149, 151, 152 92. 140, 149, 151 93. 141, 149, 150 94. 140, 145, 149, 150, 151, 152 95. 140, 150, 152 96. 136, 141 97. 138, 139, 140, 141, 142, 143 98. 138, 142 99. 142, 143, 144, 145 100. 143, 144, 146, 147 101. 137, 141 102. 138, 139, 141 103. 140, 141, 144 104. 138, 141, 147 105. 149, 152 106. 151 107. 150, 151, 152 n = 20, r = 8 108. 155, 156, 157 109. 155, 158 110. 153, 159 111. 157, 158, 159, 160
112. 158, 159 113. 114. 115. 116. 117. -118. 155, 156 119. 120. 121. -122. 154, 155, 156 123. 153, 155, 156 124. 155, 157, 158 125. 155 126. 156, 158 127. 155, 158 128. 155, 156, 157 129. 153, 158, 159 130. 153, 157, 158 131. 156 132. 155, 157, 160 133. 157, 158, 159 134. 158 n = 22, r = 9 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. -n = 24, r = 10 153. 161 154. 162 155. 163 156. 162, 163 157. 163 158. 161, 163 159. 161 160. 163 n = 28, r = 12 161. 162. 163. -Second layer n = 9, r = 4 164. 166, 167, 170
165. 167, 168, 169 n = 12, r = 6 166. 174, 176 167. 171, 172, 173, 176 168. 171, 173, 175 169. 173 170. 173, 174, 176 n = 15, r = 8 171. 177 172. 177 173. 177, 178, 180 174. 179, 180 175. 178 176. 180 n = 18, r = 10 177. 181 178. 181,182 179. 182 180. 182 n = 21, r = 12 181. -182. 183 n = 24, r = 14 183. 184 n = 27, r = 16 184. -Third layer n = 8, r = 4 185. 186 n = 12, r = 7 186. 187 n = 16, r = 10 187.
-Received
I. Barbedo, Center for Research and Development in Mathematics and Applica-tions, EsACT, Politechnic Institute of Bragan¸ca, Rua Jo˜ao Maria Sarmento Pimentel, Apartado 128, 5370-326 Mirandela, Portugal.
E-mail: [email protected]
D. M. Cardoso, Center for Research and Development in Mathematics and Applica-tions, Department of Mathematics, University of Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal.
E-mail: [email protected]
D. Cvetkovi´c, Mathematical Institute, Serbian Academy of Sciences and Arts, Knez Mihailova 26, 11 000 Belgrade, Serbia.
E-mail: [email protected]
P. Rama, Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal.
E-mail: [email protected]
S.K. Simi´c, Mathematical Institute, Serbian Academy of Sciences and Arts, Knez Mihailova 26, 11 000 Belgrade, Serbia.
